Luis plans to purchase some new baseball bats and gloves before spring training begins. Luis wants to buy the same bats and gloves he purchased last spring, but he can’t remember the price of each item. Luis recalls making two purchases last spring, each totaling $135 and both tax exempt. The first purchase was for a glove and three bats. The second purchase was for two gloves and a bat. If the current prices of these items are the same as last spring, how much will Luis pay this spring for three gloves and four bats?
If we let g and b represent the prices of a glove and a bat, respectively, we can derive two equations: g + 3b = 135 and 2g + b = 135. We want to determine the price of three gloves and four bats. Combined, the two purchases from last spring were for (g + 3b) + (2g + b) = 3g + 4b, which is three gloves and four bats. So, we can add the two totals to find that the price is 135 + 135 = $270.
Based on the information from the previous problem, how many bats can Luis purchase for the same amount he would pay for three gloves?
From the previous problem, we know that g + 3b = 135 and 2g + b = 135, so we can set the two expressions equal to one another to get the equation g + 3b = 2g + b. Simplifying, we get 2b = g. That means each glove Luis purchases costs the same amount as two bats. Therefore, for the same amount paid for three gloves, Luis could purchase 3 × 2 = 6 bats.
Luis has $300 budgeted for the purchase of new baseball bats and gloves this spring. Luis needs to purchase a minimum of two new bats and two new gloves, and he wants to spend as much as possible of the budgeted amount. Based on the previous problems, what is the greatest number of bats Luis can purchase?
In the previous problem, we determined that 2b = g. We can substitute in the equation g + 3b = 135 to get 2b + 3b = 135. Simplifying and solving, we find that the price of a bat is 5b = 135 à b = $27. That means price of a glove is 2 × 27 = $54. Let’s first determine the greatest number of bats Luis can purchase along with two gloves. The price of two gloves is 2 × 54 = $108. That leaves 300 – 108 = $192 to spend on bats. Since 192 ÷ 27 = 7 1/9, Luis can purchase seven bats for a total of 27 × 7 = $189. Luis will spend 108 + 189 = $297 to purchase two gloves and seven bats. Now, let’s see if this maximizes the amount he can spend of the $300 budget. The GCF of 27 and 54 is 27, and 300 ÷ 27 = 11 1/9. Of the $300, Luis can spend no more than 27 × 11 = $297 purchasing bats and gloves. Therefore, using as much as possible of the budget, the greatest number of bats Luis can purchase (along with at least two gloves) is 7 bats.
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